Essential self-adjointness of Schr\"odinger-type operators on manifolds

M. Braverman ; O. Milatovic ; M. A. Shubin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 4, pp. 641-692

Résumé

Several conditions are obtained ensuring the essential self-adjointness of a Schrödinger-type operator $H_V=D^*D+V$, where $D$ is a first-order elliptic differential operator acting on the space of sections of a Hermitian vector bundle $E$ over a manifold $M$ with positive smooth measure $d\mu$ and $V$ is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on $M$ naturally associated with $H_V$. The results generalize theorems of Titchmarsh, Sears, Rofe-Beketov, Oleinik, Shubin, and Lesch. It is not assumed a priori that $M$ is endowed with a complete Riemannian metric. This enables one to treat, for instance, operators acting on bounded domains in $\mathbb R^n$ with Lebesgue measure. Singular potentials $V$ are also admitted. In particular, a new self-adjointness condition is obtained for a Schrödinger operator on $\mathbb R^n$ whose potential has a Coulomb-type singularity and can tend to $-\infty$ at infinity. For the special case in which the principal symbol of $D^*D$ is scalar, more precise results are established for operators with singular potentials. The proofs of these facts are based on a refined Kato-type inequality modifying and improving a result of Hess, Schrader, and Uhlenbrock.